OK, I don't "get it" but I understand what you are saying. Which might be as far as I can get this.

These are the properties of exponents. Let me try to explain it to you again. Let denote a nonzero real number.

Let's reverse this:

See what we are doing here? We are condensing this long expression, and we do that by writing a number in superscript near the number to indicate how many times it is multiplied. To generalize this, let's say:

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Now, we got that covered. Now, what if we multiply two powers?

Think about it. If you expand it:

Don't you agree that:

To generalize this, we say:

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Moving on to the power of a power. Suppose you have:

Let's expand now:

Now, I told you what happens when we multiply two powers. This expression above becomes:

Notice that 4 appears 5 times. Thus:

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Moving on...what happens when I take the quotient of powers? Well, let me introduce to you the definition of exponents first:

Assume a and n are real numbers. Then:

Going back to the original problem: quotient of powers.

Don't you agree that:

And thus:

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Now, dealing with difference powers. It should seem obvious to you now that